The Style
It is called an epistle and so it is,but of a peculiar kind. In fact, as has been said, it begins like a treatise, proceeds like a sermon, and concludes like a letter. It is, in fact, more like a literary composition than any other New Testament book as Deissmann shows : "It points to the fact that the Epistle to the Hebrews, with its more definitely artistic, more literary language (corresponding to its more theological subject matter), constituted an epoch in the history of the new religion. Christianity is beginning to lay hands on the instruments of culture; the literary and theological period has begun." (Light From the Ancient East, pp. 70f.)
But Blass (Die Rhythmen der asianischen und romischen Kunstprosa, 1905) argues that the author of Hebrews certainly and Paul probably were students of Greek oratory and rhetoric. He is clearly wrong about Paul and probably so about the author of Hebrews. There is in Hebrews more of "a studied rhetorical periodicity (Thayer), but with many "parenthetical involutions" (Westcott) and with less of "the impetuous eloquence of Paul."
The eleventh chapter reveals a studied style and as a whole the Epistle belongs to the literary Koine rather that to the vernacular. Moulton (Cambridge Biblical Essays, p. 483) thinks that the author did not know Hebrew but follows the Septuagint throughout in his abundant use of the Old Testament.
The Author
Origen bluntly wrote: "Who wrote the Epistle God only knows certainly" as quoted by Eusebius. Origen held that the thoughts were Paul's while Clement of Rome or Luke may have written the book. Clement of Alexandria (Eusebius says) thought that Paul wrote it in Hebrew and that Luke translated it into Greek. No early writer apparently attributed the Greek text to Paul. Eusebius thought it was originally written in Hebrew whether by Paul or not and translated by Clement of Rome. But there is no certainty anywhere in the early centuries.
It was accepted first in the east and later in the west which first rejected it. But Jerome and Augustine accepted it. When the Renaissance came Erasmus had doubts, Luther attributed it to Apollos, Calvin denied the Pauline authorship. In North Africa it was attributed to Barnabas. In modern times Harnack has suggested Priscilla, but the masculine participle in 11:32 (me diegoumenon) disposes of that theory. The oldest Greek MSS. (Aleph A B) have simply Pros Hebraious as the title, but place it before the Pastoral Epistles and Philemon.
In the light of all the facte one can only make a guess without a sense of certainty. For myself I should with Luther guess Apollos as the most likely author of this book which is full of the Spirit of God.